The regularity of edge rings and matching numbers

Let $K[G]$ denote the edge ring of a finite connected simple graph $G$ on $[d]$ and $\text{mat}(G)$ the matching number of $G$. It is shown that $\reg(K[G]) \leq \mat(G)$if$G$isnon-bipartiteand$K[G]$isnormal,andthat$\reg(K[G]) \leq \mat(G) - 1$if$G$ is bipartite.